15.2.16 problem 16

Internal problem ID [2886]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 6, page 25
Problem number : 16
Date solved : Sunday, March 30, 2025 at 12:43:18 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (\frac {x}{y}+\frac {y}{x}\right ) y^{\prime }+1&=0 \end{align*}

✓ Maple. Time used: 0.286 (sec). Leaf size: 115

ode:=(x/y(x)+y(x)/x)*diff(y(x),x)+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {-c_1 \,x^{2}-\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= \frac {\sqrt {-c_1 \,x^{2}+\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {-c_1 \,x^{2}-\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {-c_1 \,x^{2}+\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.102 (sec). Leaf size: 121

ode=(x/y[x]+y[x]/x)*D[y[x],x]+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ \end{align*}

✓ Sympy. Time used: 3.618 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x/y(x) + y(x)/x)*Derivative(y(x), x) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = - \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}\right ] \]